Sensim Math · Depth 한국어

3-1 · Plane Figures

Count segments between points on a line

4.G.A.13.OA.D.9 · take · grade 4

Archetype: Systematically Count Shapes in a Figure · step in a 5-type progression

▶ Practice — 5 problems

How many line segments are there in the figure below?

The figure is a closed shape made entirely of straight lines. It is a large rectangle whose bottom edge is interrupted by an inward step-shaped notch (a battlement-like cutout), so the outline runs through a sequence of horizontal and vertical segments. Count every line segment (a straight stretch joining one corner point to the next) along the outline of the figure.

Show solution

Understand

The figure is a rectangle whose bottom edge is cut by an inward, battlement-shaped notch with a small raised tab in the middle. I must count how many straight line segments make up the whole outline, where a segment runs from one corner to the next.

Givens
  • The figure is a single closed shape made only of straight horizontal and vertical lines.
  • It is a large rectangle with an inward step-shaped (battlement) notch cut into the bottom middle.
  • The notch contains a small central tab that points upward, making a crown-like bottom profile.
Unknowns
  • The number of straight line segments along the outline of the figure.
Constraints
  • A line segment is one straight stretch between two neighboring corner points.
  • Each corner is where the outline changes direction.

Plan

#1 Draw a Diagram · also uses: #2 Make a Systematic List

The cleanest way to count segments on a rectilinear outline is to trace it like a pencil walk and mark every corner where the direction changes; for a closed rectilinear shape the number of segments equals the number of corners. Walking the outline once, in order, is a systematic list that guarantees nothing is skipped or repeated.

Execute

#1 Draw a Diagram 4.G.A.1
Start at the top-left corner and walk clockwise. The top edge is one segment (1), the right edge down is one segment (2). So far 2 segments along the rectangle's top and right.
22
Recognizing each straight side as one line segment is exactly the Grade 4 idea of identifying line segments in a figure.
#2 Make a Systematic List 4.G.A.1
Continuing clockwise across the bottom: floor on the right (3), up into the notch (4), short ledge left (5), up the right side of the central tab (6), across the top of the tab (7), down the left side of the tab (8), short ledge left (9), down the left wall of the notch (10), and the floor on the left (11). That is 9 more segments.
2+9=112 + 9 = 11
Listing each straight stretch in walking order keeps every up/down/across step accounted for exactly once.
#1 Draw a Diagram 3.OA.D.9
Finally the left edge goes up from the bottom-left corner back to the top-left corner: that is the last segment (12), which closes the outline.
11+1=1211 + 1 = 12
Counting the final closing side is simple addition, and the walk returning to its start confirms the loop is complete.
Answer: 12 line segments

Review

A plain rectangle has 4 segments. Cutting a battlement notch with a central tab into the bottom replaces the single bottom edge with a zig-zag that adds 8 extra corners, giving 12. Because the shape is closed and rectilinear, the count of segments equals the count of corners, and 12 corners is what the figure shows.

Look for a pattern (tool 5): each rectangular 'bump' or 'dip' added to a straight edge adds 2 segments. The bottom edge here has a notch plus a tab, i.e. 4 such steps beyond the plain edge, so 4 + 2x4 = 12 total, matching the trace.

Standards · min grade 4

  • 4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying each straight stretch of the outline as one line segment between corners.
  • 3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Adding the segment counts and using the bump-adds-two pattern to check the total.
💡 Walk the outline like a pencil and count every time it turns - each straight stretch between turns is one line segment!